Product Rule (Cambridge International IGCSE Further Maths)

Product rule

What is the product rule?

• The product rule is a formula that allows you to differentiate a product of two functions
• If $y=u\times v$ where u and v are functions of x then the product rule is:

$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$

• In function notation, if $f(x)=g(x)\times h(x)$ then the product rule can be written as:

$f'(x)=g(x)h'(x)+h(x)g'(x)$

• The easiest way to remember the product rule is, for $y=u\times v$ where u and v are functions of x:

$y'=uv'+vu'$

How do I know when to use the product rule?

• The product rule is used when we are trying to differentiate the product of two functions

  • These can easily be confused with composite functions (see chain rule)

    • $\mathrm{sin}(\mathrm{cos}x)$ is a composite function, “sin of cos of $x$”
    • $\mathrm{sin}x\mathrm{cos}x$ is a product, “sin x times cos $x$”

How do I use the product rule?

• Make it clear what$u,v,u'$ and$v'$ are

  • arranging them in a square can help

    • opposite diagonals match up

STEP 1

Identify the two functions,$u$ and$v$

Differentiate both$u$ and$v$ with respect to$x$ to find$u'$ and $v'$

STEP 2

Obtain $\frac{dy}{dx}$ by applying the product rule formula$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$ Simplify the answer if straightforward to do so or if the question requires a particular form

Exam Hint

• The product rule formula is not on the list of formulas page – make sure you know it

• Don't confuse the product of two functions with a composite function:

  • The product of two functions is two functions multiplied together
  • A composite function is a function of a function

• To differentiate composite functions you need to use the chain rule

Worked Example